پیشرفت‌های اخیر در چندجمله‌ای‌های متعامد، توابع خاص و کاربردهای آن‌ها: یازدهمین سمپوزیوم بین‌المللی در مورد چندجمله‌ای‌های متعامد، توابع خاص و کاربردهای آن‌ها، ۲۹ اوت تا ۲ سپتامبر ۲۰۱۱، دانشگاه کارلوس سوم مادرید، لگانه

This volume contains the proceedings of the 11th International Symposium on Orthogonal Polynomials, Special Functions, and their Applications, held August 29–September 2, 2011, at the Universidad Carlos III de Madrid in Leganés, Spain. The papers cover asymptotic properties of polynomials on curves of the complex plane, universality behavior of sequences of orthogonal polynomials for large classes of measures and its application in random matrix theory, the Riemann–Hilbert approach in the study of Padé approximation and asymptotics of orthogonal polynomials, quantum walks and CMV matrices, spectral modifications of linear functionals and their effect on the associated orthogonal polynomials, bivariate orthogonal polynomials, and optimal Riesz and logarithmic energy distribution of points. The methods used include potential theory, boundary values of analytic functions, Riemann–Hilbert analysis, and the steepest descent method. Preface 10 Life and work (so far) of Paco Marcellán 12 Life of Paco Marcellán 12 1. The early years 12 2. At the university 13 3. Doctoral dissertation 14 4. Research career 16 5. Professional career 17 6. Non-academic activities 17 Work of Paco Marcellán 19 7. Cassinian curves and lemniscates 19 8. Bar-le-Duc: the start of OPSFA 20 9. Perturbations of orthogonal polynomials 21 10. Sobolev orthogonal polynomials 22 11. Other topics 25 12. Paco as an organizer 26 13. Epilog 27 References 27 Asymptotics of L_{p}-norms of Hermite polynomials and Rényi entropy of Rydberg oscillator states 30 1. Introduction 30 2. L_{p}-norms of Hermite polynomials: Asymptotics (n→∞) 31 3. Rényi entropy of Rydberg oscillator states 37 4. Conclusions 37 Acknowledgements 38 References 38 The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere 42 1. Introduction 42 2. The logarithmic energy on the unit sphere in R3 44 3. Asymptotics of optimal Riesz s-energy 47 4. Conjectures for the Riesz s-energy 51 5. Numerical Results 54 6. Proofs 56 7. Motivations for conjectures 68 Appendix A. Auxiliary results 70 References 70 Spectral transformations of hermitian linear functionals 74 1. Introduction 74 2. Minimality 77 3. Characterization 79 4. Applications 88 Acknowledgements 92 References 92 Numerical study of higher order analogues of the Tracy–Widom distribution 94 1. Introduction 94 2. Riemann–Hilbert characterization of the kernels 98 3. Numerical study of the distributions 100 4. Plots and open problems 102 References 108 Comb functions 110 1. Introduction 110 2. Comb representation of LP entire functions 113 3. MacLane’s theorem 115 4. Representation of Green’s and Martin’s functions 119 5. More general combs 123 6. Uniform approximation and extremal problems 124 7. Spectral theory and harmonic analysis 126 References 127 Orthogonality relations for bivariate Bernstein-Szegő measures 130 1. Introduction 130 2. Preliminaries 131 3. Orthogonality relations in L2(1/|p_{n,m}|2dσ) 133 4. The proof of Theorem 3.1 135 5. The bivariate Christoffel-Darboux formula 139 6. Parametric orthogonal polynomials 140 References 142 Quantum walks and CMV matrices 144 1. Introduction and contents of the paper 144 2. Szegő polynomials and CMV matrices 145 3. Traditional quantum walks 147 4. Quantum walks resulting from a probability measure 148 5. The Verblunsky coefficients for Riesz’s measure 149 6. Building the backbone 149 7. Building up the chain 150 References 151 Discrete beta ensembles based on Gauss type quadratures 154 1. Introduction 154 2. Proof of Theorem 1.1 and Corollary 1.2 159 3. Proof of Theorem 1.3 and Corollary 1.4 164 References 174 Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later 176 1. Padé approximants to algebraic functions 176 2. Statement of the problem 179 3. Heine and Stieltjes, or asymptotic analysis based on the Liuoville-Green approximation 181 4. Case of p=3: Chebotarev compact and the Riemann surface 188 5. Riemann and Hilbert, or the non-linear steepest descent analysis 192 6. Wrapping up, or matching the asymptotic formulas and the Nuttall conjecture 201 References 202 Orthogonal Polynomials and S-curves 206 1. Complex orthogonal polynomials and S-curves 206 2. Padé approximants for functions with branch points 209 3. An existence theorem for an S-curve in harmonic external field. 211 4. Critical measures and equilibrium measures of S-curves 214 5. Rational external fields 217 6. Green’s and the vector equilibrium problems 220 7. Strong asymptotics 227 8. Generalized Jacobi polynomials 231 9. Proof of the existence theorem 234 References 247 Fast decreasing and orthogonal polynomials 252 1. Fast decreasing polynomials 252 2. Quasi-uniform zero spacing of orthogonal polynomials 254 3. Christoffel functions 255 4. Nonsymmetric fast decreasing polynomials 258 5. Fast decreasing polynomials on the complex plane 258 6. Christoffel functions on a system of Jordan curves 259 7. Universality 262 8. The Levin-Lubinsky fine zero spacing theorem 263 References 264 J. Arvesú, G. López Lagomasino, Editors. Includes Bibliographical References.